A navy suit is an excellent blank canvas. That's the beauty of it. Wear whatever color socks you feel like. To a more formal occasion, like a business interview or client meeting or what have you, I'd stick with a more conservative color. But to a party or an informal gathering, wear whatever color socks you want.

You'll get a lot of opinions here about "fun socks" and their pros and cons. But there are very few colors that absolutely will not go well with navy. Just don't wear white athletic socks or something.

As for matching, tread carefully. If you want to pick up the color of your socks in one of your accessories (tie, cufflinks, etc.), it's generally fine so long as you're not overdoing it. If you match too many items, you risk looking overly studied and/or cheesy.

Well, there's "matching" and then there's complementing. Don't wear a tie and pocket square of the exact same color. But you can wear a blue tie and a pocket square that has some blue in it, for instance. You can also wear complementary colors, like blue and red. It takes some experience and a decent eye for color to figure out which colors go well together. But you're right in that avoiding a direct match is usually a good idea.

I think red with dark brown shoes (like the ones in my sig ) could work.

I think navy suits work better with brown shoes for sure. I just had a law firm interview that day and needed to wear the navy suit, red striped tie, white shirt, and PAs. The socks were my nonconformist touch. Got the job though!